Applied and Computational Harmonic Analysis最新文献

英文中文

Anisotropic refinable functions and the tile B-splines 各向异性可细化函数和平铺b样条

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-28 DOI: 10.1016/j.acha.2024.101727

Vladimir Yu. Protasov , Tatyana Zaitseva

The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was not before 2019 that the non-isotropic case was done by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in

W_{2}^{k} (R^{n})

by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided.

可细化函数的正则性在两种情况下得到了很好的理解：1)单变量2)具有各向同性膨胀矩阵的多变量。一般（非各向同性）情况提供了很大的阻力。直到2019年，才通过开发矩阵方法完成了非各向同性的情况。本文进一步将在上述特殊情况下非常有效的Littlewood-Paley型方法推广到具有任意膨胀矩阵的一般方程。利用特殊锥上有限维线性算子的Perron特征值，给出了W2k（Rn）中高阶正则性的表达式。将这些结果应用到最近引入的平铺b样条上，证明了它们比经典的同阶平铺b样条具有更高的平滑性。此外，两位数平铺b样条在所有同阶近似的可细化函数中具有最小的掩模支持。这特别证明了相应细分方案的算法复杂度最低。给出了算例和数值结果。

{"title":"Anisotropic refinable functions and the tile B-splines","authors":"Vladimir Yu. Protasov , Tatyana Zaitseva","doi":"10.1016/j.acha.2024.101727","DOIUrl":"10.1016/j.acha.2024.101727","url":null,"abstract":"<div><div>The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was not before 2019 that the non-isotropic case was done by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101727"},"PeriodicalIF":2.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Scale dependencies and self-similar models with wavelet scattering spectra 小波散射谱的尺度依赖性和自相似模型

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-19 DOI: 10.1016/j.acha.2024.101724

Rudy Morel , Gaspar Rochette , Roberto Leonarduzzi , Jean-Philippe Bouchaud , Stéphane Mallat

Multi-scale non-Gaussian time-series having stationary increments appear in a wide range of applications, particularly in finance and physics. We introduce stochastic models that capture intermittency phenomena such as crises or bursts of activity, time reversal asymmetries, and that can be estimated from a single realization of size N. Variations at multiple scales are separated with a wavelet transform. Non-Gaussian properties appear through dependencies of wavelet coefficients across scales. We define maximum entropy models from the joint correlation across time and scales of wavelet coefficients and their modulus. Diagonal matrix approximations are estimated with a wavelet representation of this joint correlation. The resulting diagonals define

O (\log^{3} N)

moments that are called scattering spectra. A notion of wide-sense self-similarity is defined from the invariance of scattering spectra to scaling, which can be tested numerically on a single realization. We study the accuracy of maximum entropy scattering spectra models for fractional Brownian motions, Hawkes processes, multifractal random walks, as well as financial and turbulent time-series.

具有静态增量的多尺度非高斯时间序列出现在广泛的应用中，尤其是在金融和物理领域。我们引入的随机模型能捕捉间歇现象，如危机或活动爆发、时间反转不对称，并能从大小为 N 的单一实现中进行估算。非高斯特性通过跨尺度小波系数的依赖性显现出来。我们根据小波系数及其模量在时间和尺度上的联合相关性定义最大熵模型。对角矩阵近似值是用这种联合相关性的小波表示来估算的。由此得到的对角矩阵定义了 O(log3N) 矩，称为散射谱。根据散射谱对缩放的不变性，我们定义了广义自相似性的概念，并可在单个实现上对其进行数值测试。我们研究了分数布朗运动、霍克斯过程、多分形随机游走以及金融和湍流时间序列的最大熵散射谱模型的准确性。

{"title":"Scale dependencies and self-similar models with wavelet scattering spectra","authors":"Rudy Morel , Gaspar Rochette , Roberto Leonarduzzi , Jean-Philippe Bouchaud , Stéphane Mallat","doi":"10.1016/j.acha.2024.101724","DOIUrl":"10.1016/j.acha.2024.101724","url":null,"abstract":"<div><div>Multi-scale non-Gaussian time-series having stationary increments appear in a wide range of applications, particularly in finance and physics. We introduce stochastic models that capture intermittency phenomena such as crises or bursts of activity, time reversal asymmetries, and that can be estimated from a single realization of size <em>N</em>. Variations at multiple scales are separated with a wavelet transform. Non-Gaussian properties appear through dependencies of wavelet coefficients across scales. We define maximum entropy models from the joint correlation across time and scales of wavelet coefficients and their modulus. Diagonal matrix approximations are estimated with a wavelet representation of this joint correlation. The resulting diagonals define <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>⁡</mo><mi>N</mi><mo>)</mo></math></span> moments that are called scattering spectra. A notion of wide-sense self-similarity is defined from the invariance of scattering spectra to scaling, which can be tested numerically on a single realization. We study the accuracy of maximum entropy scattering spectra models for fractional Brownian motions, Hawkes processes, multifractal random walks, as well as financial and turbulent time-series.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101724"},"PeriodicalIF":2.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Multidimensional unstructured sparse recovery via eigenmatrix 通过特征矩阵进行多维非结构化稀疏恢复

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-19 DOI: 10.1016/j.acha.2024.101725

Lexing Ying

This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors proposed for the one-dimensional problems. This note extends the eigenmatrix approach to multidimensional problems, providing a rather unified treatment for general kernels and unstructured sampling grids in both real and complex settings. Numerical results are provided to demonstrate the performance of the proposed method.

本说明探讨了多维非结构稀疏恢复问题。例如傅立叶反演和稀疏解卷积。特征矩阵是一种数据驱动的构造，针对一维问题提出了所需的近似特征值和特征向量。本说明将特征矩阵方法扩展到多维问题，为真实和复杂环境中的一般核和非结构化采样网格提供了相当统一的处理方法。本文提供了数值结果，以证明所提方法的性能。

引用次数: 0

The beltway problem over orthogonal groups 正交群上的带路问题

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-15 DOI: 10.1016/j.acha.2024.101723

Tamir Bendory, Dan Edidin, Oscar Mickelin

The classical beltway problem entails recovering a set of points from their unordered pairwise distances on the circle. This problem can be viewed as a special case of the crystallographic phase retrieval problem of recovering a sparse signal from its periodic autocorrelation. Based on this interpretation, and motivated by cryo-electron microscopy, we suggest a natural generalization to orthogonal groups: recovering a sparse signal, up to an orthogonal transformation, from its autocorrelation over the orthogonal group. If the support of the signal is collision-free, we bound the number of solutions to the beltway problem over orthogonal groups, and prove that this bound is exactly one when the support of the signal is radially collision-free (i.e., the support points have distinct magnitudes). We also prove that if the pairwise products of the signal's weights are distinct, then the autocorrelation determines the signal uniquely, up to an orthogonal transformation. We conclude the paper by considering binary signals and show that in this case, the collision-free condition need not be sufficient to determine signals up to orthogonal transformation.

经典的带状线问题需要从圆周上无序的成对距离中恢复一组点。这个问题可以看作是晶体学相位检索问题的一个特例，即从周期性自相关中恢复稀疏信号。基于这一解释，并受冷冻电子显微镜的启发，我们提出了对正交群的自然概括：从正交群上的自相关中恢复稀疏信号，直至正交变换。如果信号的支撑点是无碰撞的，我们将对正交群上的带路问题解的数量进行约束，并证明当信号的支撑点是径向无碰撞的（即支撑点具有不同的大小）时，这个约束正好是一。我们还证明，如果信号权重的成对乘积是不同的，那么自相关决定了信号的唯一性，直到正交变换为止。最后，我们考虑了二进制信号，并证明在这种情况下，无碰撞条件不一定足以决定信号的正交变换。

{"title":"The beltway problem over orthogonal groups","authors":"Tamir Bendory, Dan Edidin, Oscar Mickelin","doi":"10.1016/j.acha.2024.101723","DOIUrl":"10.1016/j.acha.2024.101723","url":null,"abstract":"<div><div>The classical beltway problem entails recovering a set of points from their unordered pairwise distances on the circle. This problem can be viewed as a special case of the crystallographic phase retrieval problem of recovering a sparse signal from its periodic autocorrelation. Based on this interpretation, and motivated by cryo-electron microscopy, we suggest a natural generalization to orthogonal groups: recovering a sparse signal, up to an orthogonal transformation, from its autocorrelation over the orthogonal group. If the support of the signal is collision-free, we bound the number of solutions to the beltway problem over orthogonal groups, and prove that this bound is exactly one when the support of the signal is radially collision-free (i.e., the support points have distinct magnitudes). We also prove that if the pairwise products of the signal's weights are distinct, then the autocorrelation determines the signal uniquely, up to an orthogonal transformation. We conclude the paper by considering binary signals and show that in this case, the collision-free condition need not be sufficient to determine signals up to orthogonal transformation.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101723"},"PeriodicalIF":2.6,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On quadrature for singular integral operators with complex symmetric quadratic forms 关于具有复对称二次形式的奇异积分算子的正交性

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-13 DOI: 10.1016/j.acha.2024.101721

Jeremy Hoskins , Manas Rachh , Bowei Wu

This paper describes a trapezoidal quadrature method for the discretization of weakly singular, and hypersingular boundary integral operators with complex symmetric quadratic forms. Such integral operators naturally arise when complex coordinate methods or complexified contour methods are used for the solution of time-harmonic acoustic and electromagnetic interface problems in three dimensions. The quadrature is an extension of a locally corrected punctured trapezoidal rule in parameter space wherein the correction weights are determined by fitting moments of error in the punctured trapezoidal rule, which is known analytically in terms of the Epstein zeta function. In this work, we analyze the analytic continuation of the Epstein zeta function and the generalized Wigner limits to complex quadratic forms; this analysis is essential to apply the fitting procedure for computing the correction weights. We illustrate the high-order convergence of this approach through several numerical examples.

本文介绍了一种梯形正交方法，用于离散化具有复对称二次方形式的弱奇异和超奇异边界积分算子。当使用复坐标法或复等值线法求解三维时谐声学和电磁界面问题时，自然会出现此类积分算子。正交是局部修正的点阵梯形法则在参数空间中的扩展，其中修正权重由点阵梯形法则中的误差拟合矩决定，而误差拟合矩是通过爱泼斯坦兹塔函数解析得知的。在这项工作中，我们分析了爱泼斯坦zeta函数的解析延续和广义维格纳极限的复二次型；这一分析对于应用拟合程序计算修正权重至关重要。我们通过几个数值示例说明了这种方法的高阶收敛性。

引用次数: 0

Gaussian approximation for the moving averaged modulus wavelet transform and its variants 移动平均模小波变换的高斯近似及其变体

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-13 DOI: 10.1016/j.acha.2024.101722

Gi-Ren Liu , Yuan-Chung Sheu , Hau-Tieng Wu

The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this representation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques. The expansion allows us to obtain a lower bound for the Wasserstein distance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices. Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance between the distributions of a random vector derived from the time-scale representation and its normal counterpart. It is worth mentioning that the expansion consists of infinite Wiener chaos, and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases. We provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.

解析小波变换复数模的移动平均值为信号提供了一种稳健的时间尺度表示法，可用于较小的时间偏移和变形。在这项工作中，我们通过马利亚文微积分和组合技术，为静态高斯过程推导出了这一表示的维纳混沌扩展。通过该扩展，我们获得了两个长程依赖高斯过程的时间尺度表示之间以赫斯特指数为单位的瓦瑟斯坦距离下限。此外，我们还应用扩展建立了平滑瓦瑟斯坦距离的上界，以及由时间尺度表示得出的随机向量的分布与其正态对应物之间的科尔莫哥洛夫距离的上界。值得一提的是，扩展由无限维纳混沌组成，随着维纳混沌阶数的增加，投影系数会慢慢趋近于零。我们提供了这些分布距离的有理衰减上限，其速率取决于复小波系数振幅的非线性变换。

{"title":"Gaussian approximation for the moving averaged modulus wavelet transform and its variants","authors":"Gi-Ren Liu , Yuan-Chung Sheu , Hau-Tieng Wu","doi":"10.1016/j.acha.2024.101722","DOIUrl":"10.1016/j.acha.2024.101722","url":null,"abstract":"<div><div>The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this representation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques. The expansion allows us to obtain a lower bound for the Wasserstein distance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices. Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance between the distributions of a random vector derived from the time-scale representation and its normal counterpart. It is worth mentioning that the expansion consists of infinite Wiener chaos, and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases. We provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101722"},"PeriodicalIF":2.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142658789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Naimark-spatial families of equichordal tight fusion frames 等弦紧密融合框架的奈马克空间族

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-08 DOI: 10.1016/j.acha.2024.101720

Matthew Fickus, Benjamin R. Mayo, Cody E. Watson

An equichordal tight fusion frame (

) is a finite sequence of equi-dimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every

is a type of optimal Grassmannian code, being a way to arrange a given number of members of a Grassmannian so that the minimal chordal distance between any pair of them is as large as possible. Any nontrivial

has both a Naimark complement and spatial complement which themselves are

s. We show that taking iterated alternating Naimark and spatial complements of any

of at least five subspaces yields an infinite family of

s with pairwise distinct parameters. Generalizing a method by King, we then construct

s from difference families for finite abelian groups, and use our Naimark-spatial theory to gauge their novelty.

等弦密融合框（）是欧几里得空间的等维子空间的有限序列，它在康威、哈丁和斯隆的单数约束中达到相等。每一个都是一种最优格拉斯曼编码，是一种排列给定数量的格拉斯曼成员，使任意一对成员之间的最小弦距尽可能大的方法。我们的研究表明，对至少五个子空间中的任意子空间进行迭代交替的奈马克补集和空间补集，就能得到一个具有成对不同参数的无穷 s 族。然后，我们推广了金的一种方法，从有限无性群的差分族中构造出 s，并利用我们的奈马克空间理论来衡量它们的新颖性。

引用次数: 0

Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning 基于自动编码器的泛化误差保证非线性模型还原用于算子学习

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-30 DOI: 10.1016/j.acha.2024.101717

Hao Liu , Biraj Dahal , Rongjie Lai , Wenjing Liao

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.

科学和工程学中的许多物理过程自然是由无限维函数空间之间的算子表示的。在这种情况下，算子学习问题旨在从经验数据中提取这些物理过程，而由于数据的无限维或高维性，这一问题具有挑战性。应对这一挑战的一个不可或缺的组成部分是模型还原，它可以降低数据维度和问题规模。本文通过研究基于自动编码器的神经网络（AENet），利用低维非线性结构进行模型缩减。AENet 首先学习输入数据的潜在变量，然后学习从这些潜在变量到相应输出数据的转换。我们的数值实验验证了 AENet 准确学习非线性偏微分方程解算子的能力。此外，我们还建立了一套数理统计估计理论，分析了 AENet 的泛化误差。我们的理论框架表明，训练 AENet 的样本复杂度与建模过程的内在维度密切相关，同时也证明了 AENet 对噪声的鲁棒性。

{"title":"Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning","authors":"Hao Liu , Biraj Dahal , Rongjie Lai , Wenjing Liao","doi":"10.1016/j.acha.2024.101717","DOIUrl":"10.1016/j.acha.2024.101717","url":null,"abstract":"<div><div>Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101717"},"PeriodicalIF":2.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Unlimited sampling beyond modulo 超出模数的无限采样

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-24 DOI: 10.1016/j.acha.2024.101715

Eyar Azar , Satish Mulleti , Yonina C. Eldar

Analog-to-digital converters (ADCs) act as a bridge between the analog and digital domains. Two important attributes of any ADC are sampling rate and its dynamic range. For bandlimited signals, the sampling should be above the Nyquist rate. It is also desired that the signals' dynamic range should be within that of the ADC's; otherwise, the signal will be clipped. Nonlinear operators such as modulo or companding can be used prior to sampling to avoid clipping. To recover the true signal from the samples of the nonlinear operator, either high sampling rates are required, or strict constraints on the nonlinear operations are imposed, both of which are not desirable in practice. In this paper, we propose a generalized flexible nonlinear operator which is sampling efficient. Moreover, by carefully choosing its parameters, clipping, modulo, and companding can be seen as special cases of it. We show that bandlimited signals are uniquely identified from the nonlinear samples of the proposed operator when sampled above the Nyquist rate. Furthermore, we propose a robust algorithm to recover the true signal from the nonlinear samples. Compared to the existing methods, our approach has a lower mean-squared error for a given sampling rate, noise level, and dynamic range. Our results lead to less constrained hardware design to address the dynamic range issues while operating at the lowest rate possible.

模数转换器（ADC）是模拟域和数字域之间的桥梁。模数转换器的两个重要特性是采样率和动态范围。对于带限信号，采样率应高于奈奎斯特速率。此外，信号的动态范围也应在 ADC 的动态范围之内，否则信号将被削波。可以在采样前使用非线性运算符（如调制或编译）来避免削波。要从非线性运算器的采样中恢复真实信号，要么需要很高的采样率，要么需要对非线性运算施加严格的限制，而这两种情况在实际应用中都不可取。在本文中，我们提出了一种具有采样效率的广义灵活非线性算子。此外，通过仔细选择其参数，削波、调制和编带都可以看作是它的特例。我们的研究表明，当采样率高于奈奎斯特率时，带限信号可从所提算子的非线性采样中唯一识别出来。此外，我们还提出了一种从非线性采样中恢复真实信号的稳健算法。与现有方法相比，我们的方法在给定的采样率、噪声电平和动态范围内具有更低的均方误差。我们的研究结果使硬件设计的限制更少，从而在尽可能低的采样率下解决动态范围问题。

{"title":"Unlimited sampling beyond modulo","authors":"Eyar Azar , Satish Mulleti , Yonina C. Eldar","doi":"10.1016/j.acha.2024.101715","DOIUrl":"10.1016/j.acha.2024.101715","url":null,"abstract":"<div><div>Analog-to-digital converters (ADCs) act as a bridge between the analog and digital domains. Two important attributes of any ADC are sampling rate and its dynamic range. For bandlimited signals, the sampling should be above the Nyquist rate. It is also desired that the signals' dynamic range should be within that of the ADC's; otherwise, the signal will be clipped. Nonlinear operators such as modulo or companding can be used prior to sampling to avoid clipping. To recover the true signal from the samples of the nonlinear operator, either high sampling rates are required, or strict constraints on the nonlinear operations are imposed, both of which are not desirable in practice. In this paper, we propose a generalized flexible nonlinear operator which is sampling efficient. Moreover, by carefully choosing its parameters, clipping, modulo, and companding can be seen as special cases of it. We show that bandlimited signals are uniquely identified from the nonlinear samples of the proposed operator when sampled above the Nyquist rate. Furthermore, we propose a robust algorithm to recover the true signal from the nonlinear samples. Compared to the existing methods, our approach has a lower mean-squared error for a given sampling rate, noise level, and dynamic range. Our results lead to less constrained hardware design to address the dynamic range issues while operating at the lowest rate possible.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101715"},"PeriodicalIF":2.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Inverse problems are solvable on real number signal processing hardware 实数信号处理硬件可解决逆问题

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-24 DOI: 10.1016/j.acha.2024.101719

Holger Boche , Adalbert Fono , Gitta Kutyniok

Despite the success of Deep Learning (DL) serious reliability issues such as non-robustness persist. An interesting aspect is, whether these problems arise due to insufficient tools or fundamental limitations of DL. We study this question from the computability perspective by characterizing the limits the applied hardware imposes. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. On digital hardware, a conceptual barrier on the capabilities of DL for solving finite-dimensional inverse problems has in fact already been derived. This paper investigates the general computation framework of Blum-Shub-Smale (BSS) machines, describing the processing and storage of arbitrary real values. Although a corresponding real-world computing device does not exist, research and development towards real number computing hardware, usually referred to by “neuromorphic computing”, has increased in recent years. In this work, we show that the framework of BSS machines does enable the algorithmic solvability of finite dimensional inverse problems. Our results emphasize the influence of the considered computing model in questions of accuracy and reliability.

尽管深度学习（DL）取得了成功，但仍然存在严重的可靠性问题，如非稳健性。一个有趣的问题是，这些问题是由于工具不足还是深度学习的根本局限性造成的。我们从可计算性的角度出发，通过描述应用硬件带来的限制来研究这个问题。为此，我们将重点放在逆问题的类别上，其中尤其包括从测量中重建数据的任何任务。事实上，在数字硬件方面，已经推导出了解决有限维度逆问题的 DL 能力的概念障碍。本文研究了布卢姆-舒伯-斯马尔（BSS）机器的一般计算框架，描述了任意实值的处理和存储。虽然现实世界中并不存在相应的计算设备，但近年来针对实数计算硬件（通常称为 "神经形态计算"）的研究和开发却在不断增加。在这项工作中，我们证明了 BSS 机器框架确实能够实现有限维逆问题的算法求解。我们的结果强调了所考虑的计算模型在准确性和可靠性问题上的影响。

{"title":"Inverse problems are solvable on real number signal processing hardware","authors":"Holger Boche , Adalbert Fono , Gitta Kutyniok","doi":"10.1016/j.acha.2024.101719","DOIUrl":"10.1016/j.acha.2024.101719","url":null,"abstract":"<div><div>Despite the success of Deep Learning (DL) serious reliability issues such as non-robustness persist. An interesting aspect is, whether these problems arise due to insufficient tools or fundamental limitations of DL. We study this question from the computability perspective by characterizing the limits the applied hardware imposes. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. On digital hardware, a conceptual barrier on the capabilities of DL for solving finite-dimensional inverse problems has in fact already been derived. This paper investigates the general computation framework of Blum-Shub-Smale (BSS) machines, describing the processing and storage of arbitrary real values. Although a corresponding real-world computing device does not exist, research and development towards real number computing hardware, usually referred to by “neuromorphic computing”, has increased in recent years. In this work, we show that the framework of BSS machines does enable the algorithmic solvability of finite dimensional inverse problems. Our results emphasize the influence of the considered computing model in questions of accuracy and reliability.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101719"},"PeriodicalIF":2.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Applied and Computational Harmonic Analysis

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀